I want to download and use Tor, but something isn't working. I don't want to describe my situation or my problem in public, using IRC or on Stack Exchange. Is there any way that I can get private help from the Tor Project?
The Tor Project have "private" support options available that include E-Mail, Telephone and SMS. Further details can be found on their website. https://www.torproject.org/about/contact.html.en#support
Note however, that with E-Mail you're trusting people along the email route (including both ISPs), as Tor don't post any kind of PGP key to encrypt your message with. Telephone and SMS also have similar risks with telephone companies reading/listening to your commuication.
The Tor Project FAQ answers many common questions: https://www.torproject.org/docs/faq.html.en .
Specifically, this answer may help: https://www.torproject.org/docs/faq.html.en#GetTor . As mentioned in that answer, you may also email firstname.lastname@example.org from a gmail account (or other popular, https-supporting webmail provider which also allows for larger (25MB) attachments).
This webpage: https://www.torproject.org/about/contact.html.en has a variety of contact options for you beyond those you mention. In case you don't have access to that site; the best account to email is email@example.com . Again, doing so from an email provider with HTTPS support turned on is advisable, and you may (wisely) consider email as public as IRC in most cases.
FWIW: you can also email help-ar, help-es, help-fa, help-fr or help-zh for language-specific support in ARabic, ESpañol, FArsi, FRench or Mandarin Chinese (ZH)